{
  "schema_version": 1,
  "problem_number": "KOU-8.3",
  "title": "Regularity of the Sylow p-Subgroups of GL_n(Z/p^m) at the Class-p Boundary: New Cases of a Question of Wehrfritz",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In Problem 8.3 of the Kourovka Notebook (1982), B. A. F. Wehrfritz asked for which m and n the group P_n(Z_{p^m}) of all n×n matrices over Z/p^m with diagonal entries ≡ 1 and entries below the diagonal ≡ 0 (mod p), a Sylow p-subgroup of GL_n(Z/p^m), is regular. It is known to be regular when nm − 1 < p (by Merzlyakov's bound for the class) and when n² < p (Kolesnikov), and the answer is complete for m ≤ 2 (Yagzhev for m = 1; Kolesnikov, and Egorychev, Kolesnikov and Leontiev, for m = 2) and for m = 3 and p ≡ 1 (mod 3), where Kolesnikov and Leontiev proved irregularity for n ≥ (p+2)/3. Our result is a partial answer: P_n(Z/p^m) is regular in the boundary cases nm − 1 = p for (p, n, m) = (11,4,3), (17,6,3), (23,8,3), (29,10,3), (19,5,4), (23,6,4), (29,6,5), and P_n(Z/p^3) is irregular for n = (p+4)/3 and p ∈ {11, 17, 23, 29}; hence for every prime p ≤ 29, P_n(Z/p^3) is regular iff 3n − 1 ≤ p, and for p = 11 the answer is complete except for n = 4, m ≥ 4; the question stays open in general. In particular, for m = 3, 4, 5 the group can be regular although neither of the known sufficient conditions nm − 1 < p and n² < p holds, whereas for m ≤ 2 the condition nm − 1 < p is necessary. All new cases rest on computer-assisted proofs. The irregularity witnesses are the pair of matrices used by Kolesnikov and Leontiev, for which we compute the agemo subgroup ℧_1(⟨a,b⟩′) exactly. For regularity, the Hall–Petresco identity and the Magnus embedding reduce the question, at the boundary nm − 1 = p, to a finite computation in linear algebra over F_p on the two-dimensional subspaces of an n-dimensional space, shortened by a torus symmetry. For m = 3 we prove by hand that the boundary case n = (p+1)/3 is regular whenever S = e_21 + e_32 + ⋯ + e_{n,n−1} and any other element of the first subdiagonal, not a multiple of S, generate a Lie subalgebra of gl_n(F_p) that contains e_{n1}; we verify this condition by computer for n = 4, 6, 8, 10, which gives the new cases (23,8,3) and (29,10,3), and show that it fails for every odd n ≥ 5. Three conjectures suggested by the computations are stated. No printed decision of the new cases was found in the sources searched (zbMATH, arXiv, Crossref, OpenAlex, Semantic Scholar, web searches); a search that finds nothing is not a proof of novelty. This is an unrefereed note.",
  "result_type": "PARTIAL_RESULT",
  "categories": [
    "math.GR"
  ],
  "keywords": [
    "regular p-groups",
    "Sylow p-subgroups",
    "general linear groups over residue rings",
    "Wehrfritz problem",
    "Hall–Petresco identity",
    "agemo subgroup",
    "Magnus embedding",
    "Lie algebras of strictly triangular matrices",
    "computer-assisted proof",
    "partial answer",
    "Kourovka Notebook",
    "UnsolvedMath",
    "KOU-8.3",
    "math.GR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/kou-8-3/",
  "pdf_url": "https://eulersolve.org/papers/kou-8-3/paper.pdf?v=9d40d243a20c",
  "doi": "10.5281/zenodo.23585706",
  "zenodo_record_url": "https://zenodo.org/records/23585706",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The question stays open in general (for example P_4(Z/11^m) for m ≥ 4); computer-assisted proofs; Egorychev-Kolesnikov-Leontiev 2023 was seen only in its review; unrefereed.",
  "files": {
    "paper.pdf": {
      "sha256": "9d40d243a20c0e889438be561443bdf8027d086a7bd188f1ede6167b9129f8a1"
    },
    "source.zip": {
      "sha256": "e172bebaabd250783558680bfdd0c6ebdc216f4b449128c43ed3512c6f10e8ae"
    },
    "verification_report.md": {
      "sha256": "e691f2f9568e207a1fe59966f8a52c26855b4a5c4c7a2d2ee0033046f0f6ad13"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}